Mathematics majors at Pomona College all have to work on a year-long project which culminates in the writing of a senior thesis. This is what Pomona's mathematics department has decided will be the capstone senior experience for our major. There is a wide range of the kinds of projects that are acceptable: e.g., original research, historical exposition, expository writing on an advanced topic, etc. You can find more information about the mathematics senior theses here and here.

Below is a list of students who worked with me on their theses with brief synopses of what their work was about:

- G. Will Abele, Jr. (Pomona College Class of 2020)

**Thesis title:**Determining Causal Mechanisms with Instrumental Variable Regression Analyses

**Synopsis:**(from the introduction and the conclusion:)*[In this] thesis [I] ask the question, how does youth participation in high school athletics affect youth risky behavior? I ... focus on data from the National Longitudinal Survey of Youth 1979, a comprehensive study following the lives of a sample of American youth born between 1957 and 1964. The cohort includes 12,686 respondents (each with a corresponding ID) ages 14 to 22 when first interviewed in 1979. The data captures background influencers and reflects a conglomeration of factors, including ethnicity, gender, and economic status. Instrumental variable regression analysis ... is an extremely useful, interesting, and practical method of determining causal mechanisms in relationships and is able to forgive the pitfalls of Ordinary Least Squares. [Earlier research seems to imply that] participation in high school sports is associated with decreases in teen antisocial behavior, including physical fights and illicit drug use. However, an IV analysis shows the causal effect of high school sports participation on youth risky behavior is fairly nonexistent, except in the case of its positive direct effect on illicit drug use.*

**Area:**Data analysis. - Kimberley Jiongco Africa (Pomona College Class of 2015)

**Thesis title:**2+2 ≠ 4: Challenging Roadblocks to Access in Education

**Synopsis:**(from the abstract:)*The pathways to higher education are not always accessible to all students. The support needed to help students achieve success in education can often be found in the schools that they attend. However, there are many disparities among schools within the same district and across different communities in terms of funding, resources, teachers, counselors, and programs. This thesis first contextualizes the inequalities in education by understanding its roots. Then, it examines how students may be placed in schools through various means. This paper will unpack the mathematics behind how the decision on which school a student goes to is made and explore its limitations. We will examine this issue first as a geometric problem, then a two-sided matching problem, and finally as an optimization problem. This thesis concludes with ideas for research methodologies for mathematics with a social justice goal and directions for future research.*

**Area:**Mathematics for social justice, geometry, matching theory, optimization. - Gabrielle Andrea Badie (Pomona College Class of 2014)

**Thesis title:**Proof Formalization: History and Advancement

**Synopsis:**(from the abstract:)*By definition, a formal proof is a proof in which every logical inference has been checked all the way back to the fundamental axioms of mathematics. With this rigor in mind, computer scientists and mathematicians have developed several competing systems for doing formal proofs on a computer. These systems have their own strengths, are built on various logical foundations, and even have their own geographical biases. This paper highlights the differentiating characteristics of a select number of these popular systems, developing a birds-eye perspective of some widely used systems, with an in-depth exploration of HOL, Coq and Mizar. The QED Manifesto [5] argued that these systems are the future of mathematics, where statements and their proofs are widely available, easily searchable, and verified completely. With this in mind, we evaluate how far we are from achieving these goals, and what progress is needed with these systems to help direct researchers' efforts in developing the foundation of formal mathematics to become useful and pragmatic in a variety of fields.*

**Area:**Formal proof, history and culture of mathematics. - Anna Bessesen (Pomona College Class of 2011)

**Thesis title:**Effect of Gender in Mathematics Achievement

**Synopsis:**(from the abstract:)*The gender gap in mathematical achievement is well-documented. We set out to determine the effect of teacher gender on that gender gap. This was done by collection and examination of mathematics achievement data of students at a high-achieving college. We found a small positive correlation between a high proportion of male teachers in students' pre-collegiate work and their mathematical achievement at time of entry into Pomona College.*

**Co-advised with:**Darryl Yong of Harvey Mudd College.

**Area:**Mathematics education. - Jacob Alexander Brown (Pomona College Class of 2013)

**Thesis title:**Decomposing Poisson Space Into Symplectic Spaces

**Synopsis:**(from the abstract:)*This is an expository thesis on the paper "Primitive and Poisson Spectra of Single-Eigenvalue Twists of Polynomial Algebras" by M. Katherine Brandl. Her paper is about twisted complex polynomial rings and the relation between primitive ideals and symplectic leaves. I will give the necessary background information in algebra, geometry, and Poisson algebra needed in order to understand Brandl's paper before presenting her results.*

**Area:**Poisson algebra, Poisson geometry, algebraic geometry. - Tom Cleveland (Pomona College Class of 2012)

**Thesis title:**Gödel's Incompleteness Theorems

**Synopsis:**(from the abstract:)*In this paper I provide an introduction to the two famous incompleteness theorems of Kurt Gödel. I then go on to provide an exposition of Gödel's proof of both theorems using more accessible and updated language. I conclude with a discussion of how Gödel's results have been correctly and incorrectly applied to certain areas of study, as well as a brief mention of Gödel's own view on incompleteness.*

**Area:**Foundations of mathematics, mathematical logic, philosophy of mathematics. - Alexander Cole (Pomona College Class of 2015)

**Thesis title:**Supermanifolds

**Synopsis:**(from the abstract:)*This thesis provides a pedagogical introduction to super-mathematics in order to construct supermanifolds. Basic definitions in super-mathematics (starting from the super vector space) are given, and superspace is constructed from the Grassman algebra. The two main constructions of supermanifolds (the concrete and algebro-geometric approaches) are given and compared. Finally, we take a brief look at supersymmetry and note the connection between modern physics and super-mathematics.*

**Area:**Geometry, supermathematics. - Ian Cunningham (Pomona College Class of 2008)

**Thesis title:**Anti-Groups

**Synopsis:**(from the abstract:)*The numerous results contained in any group theory textbook are ultimately obtained from three simple properties applied to a set and a binary operation. The purpose of this paper is to explore the results of changing these initial properties and comparing the results to those of group theory. These new algebraic objects, called "anti-groups", are less innately structured than groups, but their resulting properties frequently complement the well-known properties of group theory in interesting ways.*

**Area:**Abstract algebra, group theory. - Yanai Feldman (Pomona College Class of 2019)

**Thesis title:**A Categorial Critique of Binary Thought

**Synopsis:**(from the abstract:)*Most of us do not have conscious awareness that most of the language we use to communicate our definitions, theorems, and proofs has its roots in Set Theory. Set Theory, at its core, elevates the status of things. Category Theory, on the other hand, elevates the status of relations. Major epistemological biases and metaphysical positions result as a consequence of thinking in terms of things over thinking in terms of relations. Some may tacitly assume these points of view without being aware of or appreciating how they appear in the content of our mathematics and processes involved in practicing and communicating it. In this paper, I first go over the basic machinery of Category Theory. Drawing conceptual inspiration from the Category***Set**, I then define and explore a significant species of category called a**Topos**. The paper culminates in a conceptual diagram, which aims to organize the different species of**Topoi**, and the different ways of engaging with their logic.

**Area:**Category theory, topos theory, philosophy of mathematics. - Luke Fischinger (Pomona College Class of 2016)

**Thesis title:**Quasigroups and Their Characters

**Synopsis:**(from the abstract:)*The study of finite groups and their representations is well-established territory. In this thesis, we begin by covering the basics of the theory of finite groups in order to explore some well-known concepts from group theory in the context of much less well-studied, non-associative algebraic structures called quasigroups. Upon defining quasigroups, we first explore the more inherent properties of the structure itself (e.g. the idea of subquasigroups and a generalization of LagrangeÕs theorem), before stepping into and investigating the method by which one can define a character for an algebraic structure that may lack all of the three fundamental axioms of a group (i.e. associativity, identity, and inverses).*

**Area:**Representation theory, group theory, algebraic combinatorics. - Christopher Fowler (Pomona College Class of 2012)

**Thesis title:**Supercharacter Theory and Ramanujan Sums

**Synopsis:**(from the abstract:)*This paper began as an investigation into the Supercharacter Theories of certain elementary families of finite groups. Through its progression, however, it transformed into a paper detailing how countless properties of Ramanujan sums can be explained easily and quite intuitively through the approach of supercharacters, recently developed by Diaconis-Isaacs and Andre.*

**Co-advised with:**Stephan Ramon Garcia of Pomona College.

**Area:**Representation theory, number theory. - Clara Fried (Pomona College Class of 2012)

**Thesis title:**Strategy Manipulation in Matching Theory

**Synopsis:**(from the abstract:)*In this paper, I introduce the marriage problem and discuss the strategies different players can use in order to improve their outcomes. The idea of the marriage problem is to find a one-to-one matching (marriage) between the set of men and the set of women. I show that the Deferred Acceptance Algorithm results in a stable matching that is the most preferred by the men and the least preferred by the women. The women have the incentive to manipulate their preference lists and cheat in order to improve their outcomes. I then explain that it is the dominant strategy for the men to not cheat even if they act in coalitions. However, if the men are willing to take the risk of worsening their outcome, a coalition of men can cheat in such a way that will give them a positive expectation of doing better.*

**Area:**Matching theory, game theory. - Eric Gofen (Pomona College Class of 2019)

**Thesis title:**(Re)humanizing Mathematics Education: Cultural Problem-Based Learning

**Synopsis:**(from the abstract:)*This thesis is focused on analyzing and writing K-12 math problems that aim to (re)humanize the discipline. I analyze examples of problem-based learning (PBL) problems and classrooms, then combine them with both ethnomathematical and culturally relevant and responsive problems. In each section, I use examples to identify commonalities in these different types of problems that will help math teachers write new problems for their own students.*

**Area:**Ethnomathematics, culturally relevant mathematics education, problem based learning. - Jacob A. Gómez (Pomona College Class of 2018)

**Thesis title:**Ethnomathematics and the Case for a Pedagogy of Liberation

Synopsis: (from the abstract:)*The discipline of ethnomathematics has often been relegated to discourses of multiculturalism, anthropology, and history, considered secondary to ÒtraditionalÓ and ÒrigorousÓ mathematics. Moreover, ÒacademicÓ mathematics has often been legitimized and institutionalized through its claims of universality, objectivity, and apoliticism. Debunking this last claim, I argue for a reconsidering of not only ethnomathematics, but mathematics education in general. I suggest that mathematics is taught, learned, applied, and understood better when it is contextualized as a social and cultural phenomenon. To enact a culturally conscious mathematics education, I pair ethnomathematics with the field of critical pedagogy. Arguing against a Eurocentric view of mathematics, this paper probes the political, historical, epistemological, and philosophical applications of an ethnomathematics education.*

**Area:**Ethnomathematics, mathematics education, history and culture of mathematics. - Nurullah E. Goren (Pomona College Class of 2018)

**Thesis title:**Qualifying Quantitative Literacy: Insights from Textbook Analysis

Synopsis: (from the introduction, mildly edited:)*This study explores the relationship between definitional frameworks, pedagogical practice, educational philosophies, and curriculum in quantitative literacy (QL), through a systematic comparative analysis of six commonly used QL textbooks. The end goal is to piece together the quantitative literacy puzzle, exploring how philosophy, curriculum, and educational practice interact with and influence each other in QL pedagogy.*

**Area:**Quantitative literacy, mathematics education. - Benjamin Greenberg (Pomona College Class of 2009)

**Thesis title:**Gödel's Incompleteness Theorems: Proofs and Implications

**Synopsis:**(from the abstract:)*In this paper, I explore Gödel's Incompleteness Theorems with a particular focus on how they apply to the formal theory Peano Arithmetic with Exponentiation (PE). I prove that the First Incompleteness Theorem holds for PE (PE is incomplete) using the notion of truth in PE and the fact that PE is sound, and I discuss how this proof relates to the liar paradox as well as Cantor's diagonal proof that the reals are uncountable. I then give an outline of a proof that the Second Incompleteness Theorem holds for PE (PE cannot prove its own consistency).*

**Area:**Foundations of mathematics, mathematical logic, philosophy of mathematics. - Xinyi Guo (Pomona College Class of 2012)

**Thesis title:**Group Theory in Physics

**Synopsis:**(from the abstract:)*We review basic concepts in group theory and introduce representation theory in the context of finite groups. We study the properties of irreducible vectors and operators. Applying the [theory we develop], we then work out an example in detail to demonstrate how taking advantage of the symmetry of the system can help us in making predictions [about] physical observables with great ease and elegance.*

**Area:**Representation theory, mathematical physics. - Adam Hathaway (Pomona College Class of 2017)

**Thesis title:**Algebraic Models of Logic in Quantum Mechanics

**Synopsis:**(from the abstract:)*In this paper I examine the mathematical foundations of quantum mechanics, especially with regard to the logical calculus associated with it. After showing how an algebraic structure is used to model the logic of classical physics, I will turn to the quantum case, and show how the entire mathematical theory of quantum mechanics can be derived with minimal assumptions. I then go on to examine the work that has been done to rigorously justify those assumptions that must be made to construct the theory, and how successful this program has been. I conclude by re-evaluating the algebraic structure used to model quantum mechanical systems, and discussing the philosophical implications of this topic.*

**Area:**Quantum logic, quantum mechanics, mathematical physics, philosophy. - Jordan Huard (Pomona College Class of 2020)

**Thesis title:**Algorithmically Generating Harmonic Progressions for a "New" Bebop Standard

**Synopsis:**(from the abstract:)*In this paper, we attempt to create a set of chord progressions that would provide the foundation for a new jazz piece that replicates the sound of be-bop. Prior to this, we define concepts from music theory using concepts from mathematics. In our efforts, we use Markov chain modeling and another algorithm to create new harmonic progressions. Lastly, we analyze the models from each algorithm and discuss the advantages and disadvantages of various modeling techniques.*

**Area:**Markov chain modeling, music theory. - Sophia Hui (Pomona College Class of 2019)

**Thesis title:**Leveraging Mistakes in the Mathematics Classroom

**Synopsis:**(from the abstract:)*This thesis seeks to understand and explore how students' mistakes can promote inquiry and collaborative discourse in the mathematics classroom using relevant epistemological theory in mathematics education such as humanistic mathematics, constructivism, negative knowledge, and Hungarian approach to discovery learning. There are concrete error analysis examples and tools, followed by a general framework, for educators to apply in their own mathematics classrooms.*

**Area:**Mathematics education. - Utsav Kothari (Pomona College Class of 2014)

**Thesis title:**The Theory of Coxeter Groups

**Synopsis:**(from the preface:)*Coxeter groups are abstract groups that can be defined combinatorially or algebraically. A combinatorial definition describes them most efficiently. Intuitively, they can be best understood or realized as Reflection groups using group theory and geometry. Both approaches give as valuable tools with which to study the theory of these groups to gather a fundamental understanding. In this document, I study both these definitions of Coxeter group, showing that they are equivalent, and explore some of the handy tools that enable us to understand more about Coxeter groups using each approach.*

**Area:**Abstract algebra, algebraic combinatorics. - Charles Kusi Minkah-Premo (Pomona College Class of 2016)

**Thesis title:**Motivation and Epistemological Beliefs

**Synopsis:**(from the abstract:)*This paper seeks to explore studentsÕ motivation and epistemological beliefs in different domains. The first chapter explains the authorÕs motivation for embarking on this study. The second chapter discusses motivation and epistemological beliefs and studies pertaining to both concepts in the context of psychology and education. In the third chapter, the paper offers an introduction to the statistical tool known as factor analysis which is commonly used in undertaking such investigations. The fourth chapter reviews a paper which employs factor analysis. In the fifth chapter, an outline is provided to guide future researchers who wish to undertake this study. The sixth and final chapter includes closing remarks and concludes the paper.*

**Area:**Mathematics education, statistics. - Emmanuel De Jesus Mendez (Pomona College Class of 2015)

**Thesis title:**Fermat's Last Theorem: Past Attempts & Final Proof

**Synopsis:**(from the abstract:)*In this paper, we explore the history and mathematical ideas behind one of the greatest solved math problems to date: Fermat's Last Theorem.*

**Area:**Number theory, history of mathematics. - Cesar Julian Meza (Pomona College Class of 2016)

**Thesis title:**Ethnomathematics: An Indigenous Approach

Synopsis: (from the abstract:)*The purpose of this project is to challenge and expand the Western framework of mathematical history, thinking, and applications through the study of ethnomathematics. As described by Marcia Ascher and Ubiratan DÕAmbrosio, ethnomathematics is a fusion of cultural ethnography and mathematical history focusing on the relationship between culture and mathematics created by non-Western peoples. Intricacies of this relationship can be found in various cultural practices such as language formation, food production, kinship systems, and art. This thesis explores how Aztecs of Central and Southern Mexico intertwined mathematics into their religious practices, cosmology, creation stories, and agricultural practices.*

**Area:**Ethnomathematics, mathematics education, history and culture of mathematics. - Andreea Nicolae (Pomona College Class of 2008)

**Thesis title:**The Algebra of Tropical Matrices, and their Application in Tropical Geometry

**Synopsis:**(from the abstract:)*The purpose of this thesis is to offer an introductory exposition of the applications of tropical algebra. I focus mainly [on] its applications to linear algebra and the study of matrices, paying particular attention to different methods of approaching matrix ranks.*

**Area:**Tropical geometry. - Sylvia Akueze Nwakanma (Pomona College Class of 2019)

**Thesis title:**Reading with Pictures, Picturing with Words: Probabilistic Topic Modeling of Text and Image

**Synopsis:**(from the introduction:)*Topic Modeling is a statistical technique for analyzing data and decomposing it into "topics". This thesis explores its applications in textual and visual data, building the conceptual framework of topic models on the former and extending its capabilities with the latter.*

**Area:**Textual and visual data analysis, topic modeling. - Emily Frances Proulx (Pomona College Class of 2016)

**Thesis title:**Creative Mathematical Reasoning in Assessment Tasks

**Synopsis:**(from the abstract:)*In this thesis we explore creative mathematical reasoning and how this thinking manifests in middle school level assessment tasks. We begin with an overview of the Common Core State Standards in Mathematics. We then explore characterizations of creative mathematical thinking from the early 20th century until today, providing clarity to the topic of creativity. Next, we introduce a framework for identifying creative mathematical reasoning and imitative reasoning as presented by Johan Lithner. Finally, we apply this framework to methodically selected textbooks to analyze the extent to which the Common Core State Standards in Mathematics foster creative reasoning.*

**Area:**Mathematics education, creativity. - Rina Sadun (Pomona College Class of 2014)

**Thesis title:**Game Theory and the Binding of Isaac: Modeling Motivations in the Hebrew Bible

**Synopsis:**(from the abstract:)*The application of game theory to literary works in general and the Bible in particular is promising but has not yet received the attention it deserves. Meanwhile, the biblical narrative of the Binding of Isaac (Genesis 22) is one of the most influential and deeply studied stories in the Abrahamic tradition, but continues to challenge interpreters. This paper will discuss the benefits and challenges of modeling biblical narratives, particularly the Binding of Isaac, with game theory, analyzing the models proposed by Steven J. Brams and Christina Pawlowitsch and proposing some additional ways in which the story can be modeled.*

**Area:**Game theory, Biblical studies. - Bella Senturia (Pomona College Class of 2020)

**Thesis title:**Towards a New Characterization of the Weak Order

**Synopsis:**(from the introduction:)*[T]his thesis is an exploration of the properties of standard Young tableaux and the weak order, and a discussion of the various ways that we have worked towards [a] new characterization [of the latter].*

**Area:**Algebraic combinatorics. - Courtney Sibert (Scripps College Class of 2012)

**Thesis title:**School Choice and Voucher Systems: a comparison of the drivers of educational achievement and of private school choice

**Synopsis:**(from the abstract:)*Despite promotion by well-known economists and supporting economic theory, econometric analyses of voucher systems often find that they have been unsuccessful in improving traditional measures of educational success. This paper examines a possible explanation of this phenomenon by comparing the drivers of educational achievement and of school popularity by examining private school choice. The findings of this paper indicate that there is a disconnect between school success and school popularity, which adversely effects both the demand and supply-side benefits of voucher systems. Additionally, this paper reviews matching mechanisms that seek to efficiently match students with schools based on both student and school preferences.*

**Dual thesis:**Economics and Mathematics

**Economics Advisor:**Latika Chaudhary of Scripps College.

**Area:**Economics, matching theory. - Julia Smith (Pomona College Class of 2017)

**Thesis title:**Evaluating Problem-Based Learning Mathematics Curricula, Grades 9-12

**Synopsis:**(from the abstract:)*This paper seeks to develop a new framework for qualitatively analyzing and evaluating problem-based learning (PBL) mathematics problems for grades 9-12 using relevant epistemological theory in mathematics education such as cognitive demand and intellectual need. Findings indicate that subcategories of intellectual need and levels of cognitive demand are correlated in problems across content areas.*

**Area:**Problem-based learning, mathematics education. - Jennifer Marie Stewart (Pomona College Class of 2015)

**Thesis title:**Will This Be on the Test?: Comparing Transition School Math Homework and Exams

**Synopsis:**(from the abstract:)*The Robinson CenterÕs Early Entrance Program is an accelerated education program that allows gifted young students to enter the University of Washington instead of attending high school. The Early Entrance Program uses an intensive one year academic program, called Transition School, in order to prepare its students for university coursework. In course areas like mathematics, which often builds sequentially, Transition School is crucial to ensure that students have the background they need to succeed. This year, the Center included a new online homework system, called WebAssign, in the Transition School Math curriculum. This study aims to assess how the new homework format prepares students for their exams by comparing the cognitive abilities required by both homework and test problems.*

**Area:**Mathematics education, gifted education, creativity. - Hannah Thornhill (Scripps College Class of 2016)

**Thesis title:**The Philosophy of Mathematics: A Study of Indispendability and Inconsistency

**Synopsis:**(from the introduction:)*The history of mathematics is full of mathematical problems with philosophical significance. A fundamental understanding of the nature of these problems is found by studying what mathematical objects and theorems consist of. The implications of accepting one philosophical theory over another spreads to problems in broader ontology and metaphysics. In this thesis I wish to focus on The Indispensability Argument for platonism and the possible issues it faces with the inconsistencies in the field of mathematics.*

**Dual thesis:**Philosophy and Mathematics

**Philosophy Advisor:**Yuval Avnur of Scripps College.

**Area:**Philosophy of mathematics. - Kimberly Walters (Pomona College Class of 2008)

**Thesis title:**Groebner Bases and their Applications to Integer Programming

**Synopsis:**(from the abstract:)*Linear programming problems involve optimizing a linear objective function subject to a finite set of linear constraints. If the variables must take integer values, it is called an integer program. These types of problems have many applications for practical planning and decision making in mathematics, economics, business, and engineering. This paper presents an approach to integer programming problems that utilizes the properties of Groebner bases.*

**Area:**Applied abstract algebra, operations research. - Jinglin Vannessa Wang (Pomona College Class of 2017)

**Thesis title:**Revisiting the College Admissions Problem

**Synopsis:**(from the introduction:)*Because of their wide applicability, the two-sided matching models initiated by Gale and Shapley have given rise to many interesting extensions. This thesis explores a few of them, for exposition purposes and also for solving and re-modeling the College Admissions Problem.*

**Area:**Matching theory, game theory, economic theory. - Ian Zhang (Pomona College Class of 2012)

**Thesis title:**Game-Theoretic and Search-Theoretic Models in Matching Theory

**Synopsis:**(from the abstract:)*This paper briefly surveys from a highly theoretical perspective one book and five papers on matching theory, the theory behind matching-making. In particular, we introduce and compare two models in matching theory -- a game-theoretic model and a search-theoretic model -- developed by Gale and Shapley (1962) and Adachi (2003) respectively.*

**Area:**Matching theory, game theory, economic theory. - Maria Boya Zhu (Pomona College Class of 2013)

**Thesis title:**Mathematics of Happiness

**Synopsis:**(from the abstract:)*This paper explores various ways in which happiness can be represented mathematically. Specifically, it looks at the fundamental nature of happiness by studying different mathematical approaches to modeling happiness. The three approaches presented in this paper are utility theory, dynamical models, and category theory. A fundamental quality of models is that they take complex phenomena and simplify it so that only the salient aspects of a problem or situation remain. Thus, in providing different perspectives and approaches to modeling happiness, each of these models makes provides a different conception of what it means to be happy. The purpose of this paper is to analyze these different approaches and the unique insights they provide to a much more complicated human phenomenon.*

**Area:**Utility theory, dynamical systems, category theory.

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